calculate the iterated integral. ∫ - 5 5 ∫ 0 π / 2 (y + y² cos(x)) dx dy

Answer:

$\frac{500}{3}$

Explanation:

Step1: Integrate with respect to $x$

$$\int_{0}^{\frac{\pi}{2}}(y + y^{2}\cos(x))dx=\left[yx + y^{2}\sin(x)\right]_{0}^{\frac{\pi}{2}}=y\cdot\frac{\pi}{2}+y^{2}\cdot1 - (0 + 0)=\frac{\pi}{2}y + y^{2}$$

Step2: Integrate the result with respect to $y$

$$\int_{-5}^{5}(\frac{\pi}{2}y + y^{2})dy=\int_{-5}^{5}\frac{\pi}{2}y;dy+\int_{-5}^{5}y^{2}dy$$ Since $\int_{-5}^{5}\frac{\pi}{2}y;dy = 0$ (because the integrand $\frac{\pi}{2}y$ is an odd - function), and $\int_{-5}^{5}y^{2}dy=\left[\frac{1}{3}y^{3}\right]_{-5}^{5}=\frac{1}{3}(5^{3}-(-5)^{3})=\frac{1}{3}(125 + 125)=\frac{250}{3}+\frac{250}{3}=\frac{500}{3}$